Harmonic series series#

Since the early days of development of electrical devices, engineers and scientists have paid attention to effecting design improvements for economy and efficiency. Salon, in Numerical Methods in Electromagnetism, 2000 2.1 INTRODUCTION AND HISTORICAL BACKGROUND New HST images were obtained in 2002 by a team led by Marc Buie using HSTвЂs Advanced Camera for Surveys (ACS), but the results from these observations had not been published as of late 2006. The dynamic range of albedo features across the planet detected at the FOC's resolution in both the 410 and 278 nm bandpasses exceeds 5:1. The HST images and derived maps reveal that Pluto has (1) a highly variegated surface, (2) extensive, bright, asymmetric polar regions, (3) large midlatitude and equatorial spots, and (4) possible linear features hundreds of kilometers in extent. The various images that HST obtained were combined to make blue and UV maps of the planet, such as the one shown in Fig. The 20-image HST data set is longitudinally complete and rotationally resolved and obtained at both blue and ultraviolet wavelengths. These images were made by Alan Stern, Marc Buie, and Laurence Trafton using the Faint Object Camera (FOC) of the Hubble Space Telescope. And in mid-1994, it obtained the first actual images of Pluto that revealed significant details about Pluto's surface. After HST was repaired by an astronaut crew in late-1993, its optics were good enough to resolve crude details on Pluto's surface.

Albrecht and a team of collaborators cleanly separated Pluto and Charon, but it did not reveal significant details about the surface of Pluto. In 1990, the Hubble Space Telescope imaged Pluto, but owing to its then-severe optical aberrations, these images, obtained by R. Because the two groups used different data sets and different numerical techniques, their results are complementary and serve to check one another on the Charon-facing hemisphere they share in common. Because Young and Binzel used only mutual event data, their map is limited to the hemisphere of Pluto that Charon eclipses.

Harmonic series series#

The second group, consisting of Eliot Young and Richard Binzel, of MIT and SwRI, numerically fit a spherical harmonic series to each element of a finite element grid using the Charon transit mutual event lightcurve data as the model input. The first team, led by Marc Buie of Lowell Observatory has used both mutual event lightcurves and rotational lightcurve data compiled from 1954 to 1986 to compute a complete map of Pluto. The most complete mapping products obtained from photometric data inversions (variously using rotational lightcurves and mutual event lightcurves) have been obtained by two teams. Information on the latitudinal distribution of albedo can be gained by observing the evolution of this lightcurve as Pluto moves around its orbit while the pole position remains inertially fixed, assuming, of course, that the surface albedo distribution is time invariant. 2, it can be seen that Pluto's surface must contain at least three major longitudinal provinces. Because Pluto is large enough to be essentially spherical (and indeed, mutual event and stellar occultation data show it actually is), the distinct variation in this lightcurve must be related to large-scale albedo features.ДЇrom the lightcurve in Fig. However, evidence for surface markings has been available since the mid-1950s, when lightcurve modulation was first detected. Alan Stern, in Encyclopedia of the Solar System (Second Edition), 2007 4.5 Surface Appearance and MarkingsД«ecause Pluto is less than 0.1 arcsec across as seen from Earth, its disk could not be resolved until the advent of the Hubble Space Telescope. No such guarantees can be given for conditionally convergent series, though some of the above properties remain true if only one of the series to be combined is conditionally convergent. The product series, a double series, will also converge absolutely. The limit of the product will be the product of the individual series limits. The series (as a whole) may be multiplied with another absolutely convergent series. ДЄn absolutely convergent series may be added termwise to, or subtracted termwise from, or multiplied termwise with another absolutely convergent series, and the resulting series will also be absolutely convergent.If an infinite series is absolutely convergent, the series sum is independent of the order in which the terms are added. In this connection the establishment of absolute convergence is important, because it can be proved that the terms of an absolutely convergent series may be reordered according to the familiar rules of algebra or arithmetic: We now investigate the operations that may be performed on infinite series.